LetΓbe a countable group andX a BorelΓ space with an invariant Borel probability measure µ. Let E =EΓ be the countable equivalence relation defined by. We also connect the smoothness and compressibility of EΓL2(X) to the mixing properties of the action of Γ on X. In Chapter 2 we will study the descriptive complexity of an important invariant of E, namely the full group of E.
We will show that EΓL2(X) is smooth, iEΓM ALGµ is smooth and that the non-constant part of EΓL2(X) is compressible, iEΓM ALGµ\{X,∅} is compressible. We will further show that smoothness is related to rigid factors and compressibility is related to isometric factors. At the end of this chapter, we will look at some embedding properties, retention properties and their uses.
There are two frequently used topologies that are dened onAut(X, µ), namely the uniform topology and the weak topology. The goal of this chapter is to determine the descriptive complexity of the full groups of countable Borel equivalence relations and their normalizers in the weak topology.
Upper bound of the complexity of [E] and N (E)
If E is a countable Borel equivalence relation on a standard Borel space Since [E] in the uniform topology is separable by the above theorem, we can let x be a countable dense subset {Tn}n∈N ⊆ [E] in the uniform topology. If E is a countable Borel equivalence relation on a standard Borel space
Smooth equivalence relations and the closure of [E]
If E is a countable Borel equivalence relation on a standard Borel space X with invariant probability measure µ, then [E] is closed or Σ02-hard. Denote by P(X) the set of probability measures on X, by IE ⊆ P(X) the set of E-invariant Borel probability measures on X and by. EIE ⊆ P(X) the set of E-invariant ergodic Borel probability measures on X. Farrell, Varadarajan) Let E be a countable Borel equivalence relation on a standard Borel space X.
From now on, we will use the above notation: π, Xe to denote the unique ergodic decomposition of (X, E), and F to denote the equivalent Borel relation on X, which is denoted by xF y i π(x) = π(y ). For a countable BoreE, we can divide E into periodic and aperiodic parts, E = Eaperiodic ∪Eperiodic. Let E be a countable equivalent Borel relation on a standard Borel space X with a nonatomic invariant probability measure μ.
The Descriptive Complexity of [E] and N (E)
Then we deny Xi+1 and Yi+1 to be any Borel partition of Xi such that bothXi+1∩A and Yi+1 ∩A are nonzero. For non-µ-smooth E, since the periodic part is smooth, we can assume that E is aperiodic and further, by 2.3.7, we can assume that.
It is easy to check whether the ID is a Borel isomorphism between the weak topology and this topology, so N(E) is also polishable.
Introduction
We will obtain some characterizations of the smoothness and compressibility of EL2(X) and some reducibility results. Therefore, we can canonically identify Aut(X, µ) and the group of the mer algebra preserving the automorphisms of MALGµ (see [Kechris 2], p. 118). When α is a reduction and also an injection, we say that E is a Borel embedded in F, in E vB F symbols.
Moreover, if the image of α,α(X) is F-invariant, E is said to be Borel invariant embedded in F orE viB F. We can replace Borel by another class of maps, say in a class A, and generalize the concept and notations to A-reducible, ≤A, etc. For example, for the class of continuous maps, we will denote continuous reducible, embedded, invariant embedded by ≤c, vc, vic, respectively, where c stands for continuous.
If every equivalence class of E is Gδ, then E is smooth. ii) If E admits a non-atomic ergodic measure, then E is not smooth. iii). The effect on X is said to have no rigid factor if R(Γ, X, µ) contains only constant functions. In fact, the smoothness of EΓL2(X) is strictly between X without stiff factors and non-stiff. NRF) ⇐⇒ The Γ action on X has no rigid factors.
This is equivalent to saying that the Γ-action on L20(X) has no nite-dimensional invariant subspace. If EHX is ergodic for every inner subgroup H ⊆Γ, then the Γ action on X is mildly mixed in EL20(X) is even. This follows directly from Corollary 3.2.4 and EΓAut(X,µ) vic EΓL2(X,µ). iii) Assume that the Γ effect on X is mildly mixing.
For each unique irreducible representation π of G, denote by πˆ the isomorphism class of π and by Hπ its Hilbert space. Also denote by Gˆ the dual of G, which is the countable set {ˆπ:π is an irreducible unitary representation of G}. If EΓL2(G) is smooth, then Γˆπ ·π is nite for every irreducible unitary representation π of X. Or equivalently, Γπˆ/Γπ is nite for every irreducible unitary representation π of X. 2) Suppose that Γ acts on a compact Polish space X according to isometries and X has a Borel probability measure that is invariant in every isometry.
Since Iso(X) is compact, according to the Peter-Weyl theorem πX is a direct product of irreducible finite-dimensional unitary Γ-representations, say L2(X) = Q. The stabilizer f ∈L2(X) can also be described in terms of the action of Γ on X and its full group. Recall from Corollary 3.2.14 that |Γπˆ/Γπ| < ∞ for all irreducible unitary representations, π is a necessary condition for EΓL2(G) to be smooth.
Then EΓL2(G) is not smooth and there exists a sequence of normal subgroups Ni ⊆ G such that ΓG/Ni is an infinite strictly decreasing sequence of Γ, where. Since π is a direct product of irreducible unitary representations, we can find infinitely many irreducible πi,j such that ΓG/Ni = T. The negative condition (ii) holds in every case, so EΓL2(G) is not smooth. Locally finite and not smooth) Let G be a compact abelian field group and Γ act on G with automorphisms.
If K is further abelian, then the action on Gˆ is locally nite but EΓL2(G) is not smooth. Since G is abelian, Γπˆ = Γπ for every irreducible unitary representation π and in particular the Γ-action on Gˆ is locally nite.
Compressibility
So to be mildly mixing is equivalent to saying that for any BorelΓ spaceY with ergodic non-Γ-atomic invariant Borel measure ν, the diagonal Γ action on (X×Y, µ×ν) is ergodic. And being weakly mixed is equivalent to saying that for any Borel Γ-space Y with ergodic non-Γ-atomic invariant Borel probability measure ν, the diagonal Γ action on (X×Y, µ×ν) is ergodic.( see [SW] , Glasner]). Γ action on X is weakly mixed if and only if EΓL2nc(X) is compressible;. ii) EΓL2nc(X) is compressible if and only if EΓM ALGµ\{X,∅} is compressible;. iii) EΓAut(X,µ) is compressible if and only if π(Γ) is not compact. i) If EΓL2nc(X) is not compressible, then it has aΓ-invariant ergodic Borel probability measure ν.
Since pν is a Γ-invariant ergodic probability measure onL0(X), supp pν =p(supp ν) is a compact Γ-invariant subset (see Proposition 3.3.2). Then we can find a Borel Γ-spaceY with ergodic invariant Borel probability measure ν such that EΓX×Y is not ergodic. Clearly, the image measure f∗ν is a Γ-invariant Borel probability measure for L2nc(X). ii) Since EΓM ALGµ\{X,∅} viB EΓL2nc(X), the compressibility of EΓL2nc(X) implies the compressibility of EΓM ALGµ\{X,∅}.
So EΓM ALGµ\{X,∅} is not compressible. iii) The existence of a Γ-invariant Borel probability measure corresponds to the existence of the Haar probability measure on π(Γ), so this statement is obvious. If, moreover, there is a Γ-invariant Borel probability measure µ, then we say (X, µ) is isometrizable if there is a conulline-invariant subset of X that is isometrizable. We can also extend µtoµon X0 by letting µ(A) = µ(A∩X0)for every Borel A⊆X0, such that Γ acts on(X0, d)of isometries with invariant Borel probability measure µ.
We can see if a Borel Γ space X with invariant probability measure is isometrizable from its unitary representation on L2(X). A Γ-action on X is said to have a discrete spectrum if πX is the direct sum of non-dimensional irreducible unit representations. On the other hand, suppose µ is ergodic and X has discrete spectrum; Mackey showed that the Γaction on (X, µ) is isomorphic to the left translation of some homogeneous space, in particular isometriable (see [Mackey, Furman, FK]).
Since every Γ-invariant closed subspace of L2(X) is also πX(Γ)-invariant, X has discrete spectrum by the Peter-Weyl theorem. Mackey) Let Γ be a countable set and X a faithful Borel space Γ with invariant probability measure μ. If the action Γ on X has a discrete spectrum, then Γ can be embedded in a dense subgroup of a compact group G such that the space Γ (X, µ) is isomorphic to (G/K, π(ν)), where K is a closed subgroup, ν is the normalized Haar measure on G and π :G→G/K is the natural projection. So it is easy to check that the Borel space Γ (X, µ) has no rigid factors (in the sense of Denition 3.2.8) if and only if there are no non-trivial rigid factors Γ.
Some Embedding and Containment Results
Assume that a BorelΓspace X with invariant Borel probability is properly isometrizable (see next section) with proof. Therefore, any positive denite function realized on πX is the pointwise limit of a sequence of positive denite functions realized on λX. Since suitability is preserved under subsets, we can assume that EΓX is suitable and X is µ-coordinate.e. otherwise, replace X with a non-zero invariant Borel subset X0 ⊆X, which is compatible µ|X0-a.e.).
In particular, if Γ is a free non-abelian group and X is coamenable, then the action Γ is free μ-a.e. According to a basic fact from combinatorial set theory, two elements of a free set commute if and only if they are powers of a common element, that is, γ1γ2 =γ2γ1 i γ1, γ2 ∈ hγi for some γ ∈ Γ. It is easy to check that we can uniquely write ax asax =αβnα−1, where ∈N, α is reduced, β is cyclically reduced (i.e., ββ is a reduced word) and β is not a power of insignificant of other elements, i.e. ,.
Let m be a Γ-invariant mean in λΓ/H and {px} a set of local H-invariant values in EHX (see [Kaimanovich]) such that x7→px(F) is measurable for every F ∈L∞ (X, µ).